Inside the parallelogram $ABCD$ are the circles $\gamma_1$ and $\gamma_2$, which are externally tangent at the point $K$. The circle $\gamma_1$ touches the sides $AD$ and $AB$ of the parallelogram, and the circle $\gamma_2$ touches the sides $CD$ and $CB$. Prove that the point $K$ lies on the diagonal $AC$ of the paralelogram.

We are given an acute triangle $ABC$ with $AB\neq AC$. Let $D$ be a point of $BC$ such that $DA$ is tangent to the circumcircle of $ABC$. Let $E$ and $F$ be the circumcenters of triangles $ABD$ and $ACD$, respectively, and let $M$ be the midpoints $EF$. Prove that the line tangent to the circumcircle of $AMD$ through $D$ is also tangent to the circumcircle of $ABC$. [i]Proposed by Patrik Bak, Slovakia[/i]

Let $CX, CY$ be the tangents from vertex $C$ of triangle $ABC$ to the circle passing through the midpoints of its sides. Prove that lines $XY , AB$ and the tangent to the circumcircle of $ABC$ at point $C$ concur.

Given a triangle $ABC$ with $\angle C = 90^o$. a) Construct the circle with center $C$, so that one of the tangents from $A$ to that circle is parallel to one of the tangents from $B$ to that circle. b) A circle with center $C$ has two parallel tangents passing through A and go respectively. If $AC = b$ and $BC = a$, express the radius of the circle in terms of $a$ and $b$.

Let $A,B,C,D$ be the vertices of a rhombus, let $k_1$ be the circle through $B,C$ and $D$, let $k_2$ be the circle through $A,C$ and $D$, let $k_3$ be the circle through $A,B$ and $D$, let $k_4$ be the circle through $A,B$ and $C$. Prove that the tangents to $k_1$ and $k_3$ at $B$ form the same angle as the tangents to $k_2$ and $k_4$ at $A$.

$ABCD$ is an isosceles trapezoid with $BC || AD$. A circle $\omega$ passing through $B$ and $C$ intersects the side $AB$ and the diagonal $BD$ at points $X$ and $Y$ respectively. Tangent to $\omega$ at $C$ intersects the line $AD$ at $Z$. Prove that the points $X$, $Y$, and $Z$ are collinear.

In Fig. , $PA$ and $QB$ are tangents to the circle at $A$ and $B$ respectively. The line $AB$ is extended to meet $PQ$ at $S$. Suppose that $PA = QB$. Prove that $QS = SP$. [img]https://cdn.artofproblemsolving.com/attachments/6/f/f21c0c70b37768f3e80e9ee909ef34c57635d5.png[/img]

From point $A$ to circle $\omega$ tangent $AD$ and arbitrary a secant intersecting a circle at points $B$ and $C$ (B lies between points $A$ and $C$). Prove that the circle passing through points $C$ and $D$ and touching the straight line $BD$, passes through a fixed point (other than $D$).

Let $A$ be a point lies outside circle $(O)$ and tangent lines $AB$, $AC$ of $(O)$. Consider points $D, E, M$ on $(O)$ such that $MD = ME$. The line $DE$ cuts $MB$, $MC$ at $R, S$. Take $X \in OB$, $Y \in OC$ such that $RX, SY \perp DE$. Prove that $XY \perp AM$.

The parabola $y = x^2$ intersects a circle at exactly two points $A$ and $B$. If their tangents at $A$ coincide, must their tangents at $B$ also coincide?

Let $[ABCD]$ be a parallelogram with $AB <BC$ and let $E, F$ be points on the circle that passes through $A, B$ and $C$ such that $DE$ and $DF$ are tangents to this circle. Knowing that $\angle ADE = \angle CDF$ , determine $\angle ABC$. [img]https://cdn.artofproblemsolving.com/attachments/5/e/4140b92730e9d382df49ac05ca4e8ba48332dc.png[/img]

Let $ABCD$ be a quadrilateral inscribed in circle $k$. Lines $AB$ and $CD$ intersect at point $E$ such that $AB=BE$. Let $F$ be the intersection point of tangents on circle $k$ in points $B$ and $D$, respectively. If the lines $AB$ and $DF$ are parallel, prove that $A$, $C$ and $F$ are collinear.

In acute scalene triangle $ABC$ the external angle bisector of $\angle BAC$ meet $BC$ at point $X$.Lines $l_b$ and $l_c$ which tangents of $B$ and $C$ with respect to $(ABC)$.The line pass through $X$ intersects $l_b$ and $l_c$ at points $Y$ and $Z$ respectively. Suppose $(AYB)\cap(AZC)=N$ and $l_b\cap l_c=D$. Show that $ND$ is angle bisector of $\angle YNZ$. Proposed by [i]Alireza Haghi[/i]

$C$ is a point on the semicircle diameter $AB$, between $A$ and $B$. $D$ is the foot of the perpendicular from $C$ to $AB$. The circle $K_1$ is the incircle of $ABC$, the circle $K_2$ touches $CD,DA$ and the semicircle, the circle $K_3$ touches $CD,DB$ and the semicircle. Prove that $K_1,K_2$ and $K_3$ have another common tangent apart from $AB$.

Let points $P_1, P_2, P_3$, and $P_4$ be arranged around a circle in that order. (One possible example is drawn in Diagram 1.) Next draw a line through $P_4$ parallel to $P_1P_2$, intersecting the circle again at $P_5$. (If the line happens to be tangent to the circle, we simply take $P_5 =P_4$, as in Diagram 2. In other words, we consider the second intersection to be the point of tangency again.) Repeat this process twice more, drawing a line through $P_5$ parallel to $P_2P_3$, intersecting the circle again at $P_6$, and finally drawing a line through $P_6$ parallel to $P_3P_4$, intersecting the circle again at $P_7$. Prove that $P_7$ is the same point as $P_1$. [img]https://cdn.artofproblemsolving.com/attachments/5/7/fa8c1b88f78c09c3afad2c33b50c2be4635a73.png[/img]

A village has a circular wall around it, and the wall has four gates pointing north, south, east and west. A tree stands outside the village, $16 \, \mathrm{m}$ north of the north gate, and it can be [i]just[/i] seen appearing on the horizon from a point $48 \, \mathrm{m}$ east of the south gate. What is the diamter in meters, of the wall that surrounds the village?

Let $ABCD$ be a tangential quadrilateral. Let $AB$ meet $CD$ at $E, AD$ intersect $BC$ at $F$. Two arbitrary lines through $E$ meet $AD,BC$ at $M,N, P,Q$ respectively ($M,N \in AD$, $P,Q \in BC$). Another arbitrary pair of lines through $F$ intersect $AB,CD$ at $X, Y,Z, T$ respectively ($X, Y \in AB$,$Z, T \in CD$). Suppose that $d_1, d_2$ are the second tangents from $E$ to the incircles of triangles $FXY, FZT,d_3, d_4$ are the second tangents from $F$ to the incircles of triangles $EMN,EPQ$. Prove that the four lines $d_1, d_2, d_3, d_4$ meet each other at four points and these intersections make a tangential quadrilateral. Nguyễn Văn Linh

We are given a circle $k$ and a point $A$ outside of $k$. Next we draw three lines through $A$: one secant intersecting the circle $k$ at points $B$ and $C$, and two tangents touching the circle$k$ at points $D$ and $E$. Let $F$ be the midpoint of $DE$. Show that the line $DE$ bisects the angle $\angle BFC$.

Let $ABC$ be a triangle, and $M_a, M_b, M_c$ be the midpoints of $BC, CA, AB$, respectively. Extend $M_bM_c$ so that it intersects $\odot (ABC)$ at $P$. Let $AP$ and $BC$ intersect at $Q$. Prove that the tangent at $A$ to $\odot(ABC)$ and the tangent at $P$ to $\odot (P QM_a)$ intersect on line $BC$. (Li4)

From a point $P$ outside of a circle with centre $O$, tangent segments $PA$ and $PB$ are drawn. $\frac{1}{OA^2}+\frac{1}{PA^2}=\frac{1}{16}$ then $AB=$ [list=1] [*] 4 [*] 6 [*] 8 [*] 10 [/list]

Chords $ AB$ and $ CD$ of a circle intersect at a point $ E$ inside the circle. Let $ M$ be an interior point of the segment $ EB$. The tangent line at $ E$ to the circle through $ D$, $ E$, and $ M$ intersects the lines $ BC$ and $ AC$ at $ F$ and $ G$, respectively. If \[ \frac {AM}{AB} \equal{} t, \] find $\frac {EG}{EF}$ in terms of $ t$.

The diagonals of the inscribed quadrangle $ABCD$ intersect at point $K$. Prove that the tangent at point $K$ to the circle circumscribed around the triangle $ABK$ is parallel to $CD$. (A Zaslavsky)

Let $D$ be the midpoint of the hypotenuse $AB$ of a right triangle $ABC$. Let $O_1$ and $O_2$ be the circumcenters of the $ADC$ and $DBC$ triangles, respectively. a) Prove that $\angle O_1DO_2$ is right. b) Prove that $AB$ is tangent to the circle of diameter $O_1O_2$ .

Let $\ell$ be a tangent to the incircle of triangle $ABC$. Let $\ell_a,\ell_b$ and $\ell_c$ be the respective images of $\ell$ under reflection across the exterior bisector of $\angle A,\angle B$ and $\angle C$. Prove that the triangle formed by these lines is congruent to $ABC$.

The medians $AM$ and $BN$ of a triangle $ABC$ are the diameters of the circles $\omega_1$ and $\omega_2$. If $\omega_1$ touches the altitude $CH$, prove that $\omega_2$ also touches $CH$. I. Gorodnin